Tilted raw valuation-prefix rate for an arbitrary fixed coefficient box

Write $L_n=\log n$, $y_n=n^{2/9}$, $Y_n=\lfloor y_n\rfloor$, and $\mathcal P_{n,W}=\{p\text{ prime}:W<p\le Y_n\}$. For a head pattern $P$, let $S_P(A,C;n)$ be the positive $Y_n$-smooth integers in $(\lfloor An\rfloor,\lfloor Cn\rfloor]$ having exactly the valuations prescribed by $P$. For coefficients $\eta$, the tilted cell law assigns mass proportional to $\exp(L_n^{-1}\sum_{q\in\mathcal P_{n,W}}\eta_q v_q(m))$. Fix $H$, $0<A<C$ with $C>0$, $B\ge0$, and $W>1$ containing the head primes. There are $\epsilon_n$ and $N_0$ with $\epsilon_n\log L_n\to0$ such that for $n\ge N_0$, all $\lfloor An\rfloor<k\le\lfloor Cn\rfloor$, $p\in\mathcal P_{n,W}$, and band coefficients $|\eta_q|\le B$, every nonempty cell has

This supplies the raw moving-prefix terminal estimate without a smallness condition linking the coefficient box to the cutoff.